Ice Incineration

A country’s government wants to reduce the number of cars using internal combustion engines by encouraging the purchase of electric vehicles.

The total number of cars using internal combustion engines in a country at time t can be modelled by the equation

\[N\left( t \right) = \left( 3.5 \times 10^{7} + E \right)e^{- \frac{1}{10}t} – E\]

Where N(t) is the number of cars using internal combustion engines at time t years after the program was introduced and E is the number of electric vehicles on the roads before the government initiative.

Using the equation for this model,

a) Explain the significance of the number 3.5 x 107.

[1 mark]

 

b) Explain why the initial number of cars using internal combustion engines is independent of the initial number of electric vehicles.

[2 marks]

c) Calculate the number of cars using internal combustion engines after 10 years if the initial number of electric vehicles is 2 x 106.

[2 marks]

d) Analyse the suitability of this model by evaluating it for large values of t.

[2 marks]

Shrinking Species

Since 1800, the number of amphibian species, N, has been decreasing over time, t. 

A simple model shows that the rate of decrease of the number of species is proportional to the remaining number of species.

Given that the initial number of amphibian species is N0, and t is the number of years since 1800,

a) Show that \(N = N_{0}e^{- kt}\)

[4 marks]

In 2000 the number of amphibian species is 0.9N₀.

b) Find the exact value of k.

[3 marks]

c) Using the model, in what year will 20% of amphibian species be
extinct?

[3 marks]

The Swelling Sahara

Human-induced global warming is causing deserts such as the Sahara to increase in surface area.

In 1950 the area of the Sahara Desert was 9,200,000 km2, whereas in 2000 the area of the Sahara Desert had increased to 9,930,000 km2 due to human-induced global warming.

A model could be used to relate the surface area of the Sahara Desert, S km2, to the time, t, in years since 1950.

a) By first forming an exponential model for the surface area of the Sahara Desert relating S and t, show that the increase in the surface area of the Sahara Desert is approximately 0.15% per year.

[5 marks]

b) Use the model formed in (a) to estimate the size of the Sahara Desert in 2050. Give your answer in km2 to 3 s.f.

[1 mark]

A Shrinking Rainforest

  1. In a simple model, the surface area, S km2, of a shrinking rainforest depends on the time, t, in years since 1980.

The following information is available for rainforest A.

  • its surface area in 1980 was 300,000 km2
  • its surface area in 1981 was 294,000 km2
a) Use an exponential model to form, for rainforest A, a possible equation linking S with t.

[4 marks]

The surface area of rainforest A is monitored over a 30-year period. Its surface area after 30 years is 150,000 km2.

b) Evaluate the reliability of your model in light of this information.

[2 marks]

The following information is known about rainforest B.

  • it had the same surface area, in 1980, as rainforest A
  • it is harder to access by road, so the rate of deforestation is less than rainforest B and its surface area decreases more slowly than that of rainforest A
c) Explain how you would adapt the equation found in (a) so that it could be used to model the surface area of rainforest B.

[1 mark]

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